Characteristic-potentially characteristic subgroup - Revision history

Vipul: /* Formalisms */

2022-05-02T04:59:38Z

Formalisms

← Older revision		Revision as of 04:59, 2 May 2022
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	Given two subgroup properties <math>p</math> and <math>q</math>, the [[upper-hook operator]] of <math>p</math> and <math>q</math> is defined as the following property <math>r</math>: a subgroup <math>H</math> of a group <math>K</math> has property <math>r</math> if there exists a group <math>G</math> containing <math>K</math> such that <math>H</math> has property <math>p</math> in <math>G</math> and <math>K</math> has property <math>q</math> in <math>G</math>.		Given two subgroup properties <math>p</math> and <math>q</math>, the [[upper-hook operator]] of <math>p</math> and <math>q</math> is defined as the following property <math>r</math>: a subgroup <math>H</math> of a group <math>K</math> has property <math>r</math> if there exists a group <math>G</math> containing <math>K</math> such that <math>H</math> has property <math>p</math> in <math>G</math> and <math>K</math> has property <math>q</math> in <math>G</math>.

	The property of being ~~strongly~~ potentially characteristic is thus obtained by applying the upper-hook operator to the property [[characteristic subgroup]] with itself.		The property of being characteristic-potentially characteristic is thus obtained by applying the upper-hook operator to the property [[characteristic subgroup]] with itself.

	==Relation with other properties==		==Relation with other properties==

Vipul: /* Weaker properties */

2022-05-02T04:58:58Z

Weaker properties

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	* [[Stronger than::Normal-potentially relatively characteristic subgroup]]		* [[Stronger than::Normal-potentially relatively characteristic subgroup]]
	* [[Stronger than::Normal subgroup]]: {{proofofstrictimplicationat\|[[Characteristic-potentially characteristic implies normal]]\|[[Normal not implies characteristic-potentially characteristic]]}}		* [[Stronger than::Normal subgroup]]: {{proofofstrictimplicationat\|[[Characteristic-potentially characteristic implies normal]]\|[[Normal not implies characteristic-potentially characteristic]]}}
	* '''Potentially relatively characteristic subgroup''': This is a weaker notion, that turns out to be the same as normality. {{proofat\|[[Potentially relatively characteristic equals normal]]}}
	* [[Stronger than::Characteristic-extensible automorphism-invariant subgroup]]		* [[Stronger than::Characteristic-extensible automorphism-invariant subgroup]]
	* [[Stronger than::Normal-extensible automorphism-invariant subgroup]]		* [[Stronger than::Normal-extensible automorphism-invariant subgroup]]

Vipul: /* Weaker properties */

2022-05-02T04:58:47Z

Weaker properties

← Older revision		Revision as of 04:58, 2 May 2022
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	* [[Stronger than::Normal-potentially characteristic subgroup]]		* [[Stronger than::Normal-potentially characteristic subgroup]]
	* [[Stronger than::Normal-potentially relatively characteristic subgroup]]		* [[Stronger than::Normal-potentially relatively characteristic subgroup]]
	* [[Stronger than::Potentially characteristic subgroup]]: {{proofofstrictimplicationat\|[[Strongly potentially characteristic implies potentially characteristic]]\|[[Potentially characteristic not implies strongly potentially characteristic]]}}
	* [[Stronger than::Normal subgroup]]: {{proofofstrictimplicationat\|[[Characteristic-potentially characteristic implies normal]]\|[[Normal not implies characteristic-potentially characteristic]]}}		* [[Stronger than::Normal subgroup]]: {{proofofstrictimplicationat\|[[Characteristic-potentially characteristic implies normal]]\|[[Normal not implies characteristic-potentially characteristic]]}}
	* '''Potentially relatively characteristic subgroup''': This is a weaker notion, that turns out to be the same as normality. {{proofat\|[[Potentially relatively characteristic equals normal]]}}		* '''Potentially relatively characteristic subgroup''': This is a weaker notion, that turns out to be the same as normality. {{proofat\|[[Potentially relatively characteristic equals normal]]}}

Vipul at 18:28, 30 May 2009

2009-05-30T18:28:11Z

← Older revision		Revision as of 18:28, 30 May 2009
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	===Symbol-free definition===		===Symbol-free definition===

	A subgroup of a group is termed '''~~strongly~~ potentially characteristic''' if there is an embedding of the bigger group in some group such that, in that embedding both the group and the subgroup become characteristic.		A subgroup of a group is termed '''characteristic-potentially characteristic''' if there is an embedding of the bigger group in some group such that, in that embedding both the group and the subgroup become characteristic.

	===Definition with symbols===		===Definition with symbols===

	A subgroup <math>H</math> of a group <math>G</math> is termed ''potentially characteristic'' in <math>G</math> if there exists a group <math>K</math> containing <math>G</math> such that both <math>H</math> and <math>G</math> are characteristic in <math>K</math>.		A subgroup <math>H</math> of a group <math>G</math> is termed '''characteristic-potentially characteristic''' in <math>G</math> if there exists a group <math>K</math> containing <math>G</math> such that both <math>H</math> and <math>G</math> are characteristic in <math>K</math>.

	==Formalisms==		==Formalisms==
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	===Weaker properties===		===Weaker properties===

	* [[Stronger than::~~Semi~~-~~strongly~~ potentially characteristic subgroup]]		* [[Stronger than::Normal-potentially characteristic subgroup]]
	* [[Stronger than::~~Semi~~-~~strongly~~ potentially relatively characteristic subgroup]]		* [[Stronger than::Normal-potentially relatively characteristic subgroup]]
	* [[Stronger than::Potentially characteristic subgroup]]: {{proofofstrictimplicationat\|[[Strongly potentially characteristic implies potentially characteristic]]\|[[Potentially characteristic not implies strongly potentially characteristic]]}}		* [[Stronger than::Potentially characteristic subgroup]]: {{proofofstrictimplicationat\|[[Strongly potentially characteristic implies potentially characteristic]]\|[[Potentially characteristic not implies strongly potentially characteristic]]}}
	* [[Stronger than::Normal subgroup]]: {{proofofstrictimplicationat\|[[~~Strongly~~ potentially characteristic implies normal]]\|[[Normal not implies ~~strongly~~ potentially characteristic]]}}		* [[Stronger than::Normal subgroup]]: {{proofofstrictimplicationat\|[[Characteristic-potentially characteristic implies normal]]\|[[Normal not implies characteristic-potentially characteristic]]}}
	* '''Potentially relatively characteristic subgroup''': This is a weaker notion, that turns out to be the same as normality. {{proofat\|[[Potentially relatively characteristic equals normal]]}}		* '''Potentially relatively characteristic subgroup''': This is a weaker notion, that turns out to be the same as normality. {{proofat\|[[Potentially relatively characteristic equals normal]]}}
	* [[Stronger than::Characteristic-extensible automorphism-invariant subgroup]]		* [[Stronger than::Characteristic-extensible automorphism-invariant subgroup]]
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	{{fillin}}		{{fillin}}

	~~==Property operators==~~

	~~===Left transiter===~~

	~~Every characteristic subgroup of a strongly potentially characteristic subgroup is strongly potentially characteristic. In fact, the same supergroup works.~~

	That is, suppose <math>M \le G \le H</math> with <math>M</math> characteristic in <math>G</math> and <math>G</math> strongly potentially characteristic in <math>H</math>. Then, there exists a group <math>K</math> containing <math>H</math> such that both <math>G</math> and <math>H</math> are [[characteristic subgroup\|characteristic]] in <math>K</math>. Then, we also have that both <math>M</math> and <math>H</math> are characteristic in <math>K</math>, and hence <math>M</math> is strongly potentially characteristic in <math>H</math>.

Vipul: moved Strongly potentially characteristic subgroup to Characteristic-potentially characteristic subgroup

2009-05-30T18:24:10Z

moved Strongly potentially characteristic subgroup to Characteristic-potentially characteristic subgroup

← Older revision	Revision as of 18:24, 30 May 2009
(No difference)

Vipul: /* Relation with other properties */

2009-05-30T14:56:06Z

Relation with other properties

← Older revision		Revision as of 14:56, 30 May 2009
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	* [[Stronger than::Semi-strongly potentially characteristic subgroup]]		* [[Stronger than::Semi-strongly potentially characteristic subgroup]]
			* [[Stronger than::Semi-strongly potentially relatively characteristic subgroup]]
	* [[Stronger than::Potentially characteristic subgroup]]: {{proofofstrictimplicationat\|[[Strongly potentially characteristic implies potentially characteristic]]\|[[Potentially characteristic not implies strongly potentially characteristic]]}}		* [[Stronger than::Potentially characteristic subgroup]]: {{proofofstrictimplicationat\|[[Strongly potentially characteristic implies potentially characteristic]]\|[[Potentially characteristic not implies strongly potentially characteristic]]}}
	* [[Stronger than::Normal subgroup]]: {{proofofstrictimplicationat\|[[Strongly potentially characteristic implies normal]]\|[[Normal not implies strongly potentially characteristic]]}}		* [[Stronger than::Normal subgroup]]: {{proofofstrictimplicationat\|[[Strongly potentially characteristic implies normal]]\|[[Normal not implies strongly potentially characteristic]]}}
	* '''Potentially relatively characteristic subgroup''': This is a weaker notion, that turns out to be the same as normality. {{proofat\|[[Potentially relatively characteristic equals normal]]}}		* '''Potentially relatively characteristic subgroup''': This is a weaker notion, that turns out to be the same as normality. {{proofat\|[[Potentially relatively characteristic equals normal]]}}
			* [[Stronger than::Characteristic-extensible automorphism-invariant subgroup]]
			* [[Stronger than::Normal-extensible automorphism-invariant subgroup]]

	==Metaproperties==		==Metaproperties==

Vipul at 14:51, 30 May 2009

2009-05-30T14:51:20Z

← Older revision		Revision as of 14:51, 30 May 2009
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	{{subgroup property}}		{{subgroup property}}
	{{variation of\|characteristicity}}		{{variation of\|characteristicity}}
			{{termrelatedto\|NPC conjecture}}
	==Definition==		==Definition==

Vipul at 00:05, 29 May 2009

2009-05-29T00:05:15Z

← Older revision		Revision as of 00:05, 29 May 2009
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	A subgroup <math>H</math> of a group <math>G</math> is termed ''potentially characteristic'' in <math>G</math> if there exists a group <math>K</math> containing <math>G</math> such that both <math>H</math> and <math>G</math> are characteristic in <math>K</math>.		A subgroup <math>H</math> of a group <math>G</math> is termed ''potentially characteristic'' in <math>G</math> if there exists a group <math>K</math> containing <math>G</math> such that both <math>H</math> and <math>G</math> are characteristic in <math>K</math>.

	==~~=In terms of the strongly potentially operator=~~==		==Formalisms==

	~~The~~ [[subgroup property]] of being ~~'''~~potentially characteristic~~'''~~ is obtained by applying the ~~[[strongly potentially~~ operator]] to the ~~subgroup~~ property ~~of being~~ [[characteristic subgroup~~\|characteristic~~]].		{{wikilocal-section}}

			===In terms of the upper-hook operator===

			Given two subgroup properties <math>p</math> and <math>q</math>, the [[upper-hook operator]] of <math>p</math> and <math>q</math> is defined as the following property <math>r</math>: a subgroup <math>H</math> of a group <math>K</math> has property <math>r</math> if there exists a group <math>G</math> containing <math>K</math> such that <math>H</math> has property <math>p</math> in <math>G</math> and <math>K</math> has property <math>q</math> in <math>G</math>.

			The property of being strongly potentially characteristic is thus obtained by applying the upper-hook operator to the property [[characteristic subgroup]] with itself.

	==Relation with other properties==		==Relation with other properties==

Vipul: /* Relation with other properties */

2009-04-25T16:25:00Z

Relation with other properties

← Older revision		Revision as of 16:25, 25 April 2009
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	* [[Stronger than::Semi-strongly potentially characteristic subgroup]]		* [[Stronger than::Semi-strongly potentially characteristic subgroup]]
	* [[Stronger than::Potentially characteristic subgroup]]: {{proofofstrictimplicationat\|[[Strongly potentially characteristic implies potentially characteristic]]\|[[Potentially characteristic not implies strongly potentially characteristic]]}}		* [[Stronger than::Potentially characteristic subgroup]]: {{proofofstrictimplicationat\|[[Strongly potentially characteristic implies potentially characteristic]]\|[[Potentially characteristic not implies strongly potentially characteristic]]}}
	* [[Stronger than::Potentially relatively characteristic subgroup]]
	* [[Stronger than::Normal subgroup]]: {{proofofstrictimplicationat\|[[Strongly potentially characteristic implies normal]]\|[[Normal not implies strongly potentially characteristic]]}}		* [[Stronger than::Normal subgroup]]: {{proofofstrictimplicationat\|[[Strongly potentially characteristic implies normal]]\|[[Normal not implies strongly potentially characteristic]]}}
			* '''Potentially relatively characteristic subgroup''': This is a weaker notion, that turns out to be the same as normality. {{proofat\|[[Potentially relatively characteristic equals normal]]}}

	==Metaproperties==		==Metaproperties==

Vipul: /* Relation with other properties */

2009-04-23T13:04:51Z

Relation with other properties

← Older revision		Revision as of 13:04, 23 April 2009
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	* [[Stronger than::Semi-strongly potentially characteristic subgroup]]		* [[Stronger than::Semi-strongly potentially characteristic subgroup]]
	* [[Stronger than::Potentially characteristic subgroup]]		* [[Stronger than::Potentially characteristic subgroup]]: {{proofofstrictimplicationat\|[[Strongly potentially characteristic implies potentially characteristic]]\|[[Potentially characteristic not implies strongly potentially characteristic]]}}
	* [[Stronger than::Potentially relatively characteristic subgroup]]		* [[Stronger than::Potentially relatively characteristic subgroup]]
	* [[Stronger than::Normal subgroup]]		* [[Stronger than::Normal subgroup]]: {{proofofstrictimplicationat\|[[Strongly potentially characteristic implies normal]]\|[[Normal not implies strongly potentially characteristic]]}}

	==Metaproperties==		==Metaproperties==